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+--- Thread: Line Segment - Thought Experiment (/thread-25147.html)
Line Segment - Thought Experiment - BAndrew - 04-23-2014
Imagine a line segment of 1 unit(s) long (meters,feet,whatever you want).
You remove 1/3 of the segment located in the middle and you are left with 1/3 in the left and 1/3 in the right. Do this again for each of the two pieces of the segment left. Now you should have 4 smaller segments. Continue this process infinitely many times (I mean imagine it). The process is illustrated bellow:
The thing is. You obviously have removed infinetely many points (points are considered 0 dimensional), but how many points are left? Moreover, how much length of the original segment is left in the end?
RE: Line Segment - Thought Experiment - Froge - 04-23-2014
1 - (1/3) - 2(1/9) - 4(1/27) - 8(1/81)....
= 1 - (2^0)(1/3) - (2^1)(1/3^2) - (2^2)(1/3^3) - (2^3)(1/3^4) - ....
= 1 - Sigma (n=0 -> inf) 2^n / (3^n+1)
= 1 - Sigma (n=0 -> inf) (1/3) (2/3)^n
Consider just the Sigma for a moment. The formula for summation of infinite geometric series is Sum = u1 / (1 - r) where u1 is the initial term and r is the ratio between each terms.
In this series, u1 = 1/3 and r = 2/3, so Sum = (1/3) / (1 - 2/3) = (1/3) / (1/3) = 1
Substitution back into the equation, the the answer is 1 - 1 = 0
None of the segment is left.
RE: Line Segment - Thought Experiment - BAndrew - 04-23-2014
@Froge
WOW! I actually wasn't expecting that response, but you are absolutely right! This is the so called cantor set.
Cantor Set
RE: Line Segment - Thought Experiment - Froge - 04-23-2014
free reps pls
RE: Line Segment - Thought Experiment - Statyk - 04-24-2014
I was skimming the Cantor Set page and this guy's name just killed me: "Henry John Stephen Smith"
It's like... How generic can your name get? lol
(no offense, I just find that funny)